We use a Sequential Monte Carlo (SMC, [1]}) algorithm to sample from the posterior. SMC relies on importance sampling principles. This method requires us to define a sequence of distributions \(\pi _t\)  that creates a smooth transition between an initial known distribution and the posterior. The usual method is to use a tempering of the form \(\pi _t(x) \propto \pi (x \mid D)^{1/t} \)  or \(\pi _t(x) \propto \pi (x) L(D\mid x)^t \) .
This strategy is impractical in our case, as our mutation kernel makes use of a Gibbs sampler, disallowing such tempered intermediate distributions; none of our attempts in this direction were fruitful.
Instead, we use the noise parameter as a tempering parameter. For a high value of \(\nu \) , on each branch the noise matrix \(S_{noise}\)  is close to the matrix \( (\pi _0, \dots , \pi _0) \) , that is as if the model was a forest of trees constituted of only one leaf. Conversely, for \(\nu \)  small we retrieve the model we are looking for.
